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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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=Just Intonation explained=  
=Just Intonation explained=  
"Just Intonation", as we find it commonly used today, describes [[Gallery of Just Intervals|intervals]] between pitches by specifying ratios (of &lt;span style="background-color: initial;"&gt;[[http://en.wikipedia.org/wiki/Rational_number|rational numbers]]&lt;/span&gt;) between the frequencies of pitches.
Just Intonation is sometimes distinguished from //rational intonation,// by requiring that the ratios be ones of low complexity (as for example measured by [[Tenney height]]) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice, it can hardly be used except as a rank two 7-limit [[Microtempering|microtempering]] system because of certain very accurate approximations to the octave and to seven limit intervals: (6/5)^2/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)^9 is less than a cent short of an octave, while (27/25)^2 is almost precisely 7/6, the [[http://en.wikipedia.org/wiki/Septimal_minor_third|septimal minor third]].
In languages other than English, the original conceptions of "Just Intonation" are more obviously retained in the terms used in those languages: Reine Stimmung (pure, that is, beatless, tuning) in German, Натуральний стрій in Ukrainian and Gamme naturelle in French, (both referring to the "natural gamut", that is, intervals derived from the harmonic partials), Intonazione naturale (natural intonation, once again intervals derived harmonic partials) in Italian, and so on.  
In languages other than English, the original conceptions of "Just Intonation" are more obviously retained in the terms used in those languages: Reine Stimmung (pure, that is, beatless, tuning) in German, Натуральний стрій in Ukrainian and Gamme naturelle in French, (both referring to the "natural gamut", that is, intervals derived from the harmonic partials), Intonazione naturale (natural intonation, once again intervals derived harmonic partials) in Italian, and so on.  


In the English language, the term "just" referred to "true, correct", and is still used today in this sense, in the crafts. To "justify" a line of type is to fit it precisely to a certain measure, for example. The original sense, then, was similar to that sense which is clearly retained in other languages as "natural". "Just Intonation", as we find it commonly used today, describes [[Gallery of Just Intervals|intervals]] between pitches by specifying ratios (of &lt;span style="background-color: initial;"&gt;[[http://en.wikipedia.org/wiki/Rational_number|rational numbers]]&lt;/span&gt;) between the frequencies of pitches.
In the English language, the term "just" referred to "true, correct", and is still used today in this sense, in the crafts. To "justify" a line of type is to fit it precisely to a certain measure, for example. The original sense, then, was similar to that sense which is clearly retained in other languages as "natural".  
 
Just Intonation is sometimes distinguished from //rational intonation,// by requiring that the ratios be ones of low complexity (as for example measured by [[Tenney height]]) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice, it can hardly be used except as a rank two 7-limit [[Microtempering|microtempering]] system because of certain very accurate approximations to the octave and to seven limit intervals: (6/5)^2/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)^9 is less than a cent short of an octave, while (27/25)^2 is almost precisely 7/6, the [[http://en.wikipedia.org/wiki/Septimal_minor_third|septimal minor third]].


If you are used to speaking only in note names, you may need to study the relation between frequency and [[http://en.wikipedia.org/wiki/Pitch_%28music%29|pitch]]. Kyle Gann's //[[http://www.kylegann.com/tuning.html|Just Intonation Explained]]// is one good reference. A transparent illustration and one of just intonation's acoustic bases is the [[OverToneSeries|harmonic series]].
If you are used to speaking only in note names, you may need to study the relation between frequency and [[http://en.wikipedia.org/wiki/Pitch_%28music%29|pitch]]. Kyle Gann's //[[http://www.kylegann.com/tuning.html|Just Intonation Explained]]// is one good reference. A transparent illustration and one of just intonation's acoustic bases is the [[OverToneSeries|harmonic series]].
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&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Just Intonation explained"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Just Intonation explained&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Just Intonation explained"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Just Intonation explained&lt;/h1&gt;
  In languages other than English, the original conceptions of &amp;quot;Just Intonation&amp;quot; are more obviously retained in the terms used in those languages: Reine Stimmung (pure, that is, beatless, tuning) in German, Натуральний стрій in Ukrainian and Gamme naturelle in French, (both referring to the &amp;quot;natural gamut&amp;quot;, that is, intervals derived from the harmonic partials), Intonazione naturale (natural intonation, once again intervals derived harmonic partials) in Italian, and so on. &lt;br /&gt;
  &amp;quot;Just Intonation&amp;quot;, as we find it commonly used today, describes &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;intervals&lt;/a&gt; between pitches by specifying ratios (of &lt;span style="background-color: initial;"&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_number" rel="nofollow"&gt;rational numbers&lt;/a&gt;&lt;/span&gt;) between the frequencies of pitches. &lt;br /&gt;
&lt;br /&gt;
Just Intonation is sometimes distinguished from &lt;em&gt;rational intonation,&lt;/em&gt; by requiring that the ratios be ones of low complexity (as for example measured by &lt;a class="wiki_link" href="/Tenney%20height"&gt;Tenney height&lt;/a&gt;) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice, it can hardly be used except as a rank two 7-limit &lt;a class="wiki_link" href="/Microtempering"&gt;microtempering&lt;/a&gt; system because of certain very accurate approximations to the octave and to seven limit intervals: (6/5)^2/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)^9 is less than a cent short of an octave, while (27/25)^2 is almost precisely 7/6, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_minor_third" rel="nofollow"&gt;septimal minor third&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
In the English language, the term &amp;quot;just&amp;quot; referred to &amp;quot;true, correct&amp;quot;, and is still used today in this sense, in the crafts. To &amp;quot;justify&amp;quot; a line of type is to fit it precisely to a certain measure, for example. The original sense, then, was similar to that sense which is clearly retained in other languages as &amp;quot;natural&amp;quot;. &amp;quot;Just Intonation&amp;quot;, as we find it commonly used today, describes &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;intervals&lt;/a&gt; between pitches by specifying ratios (of &lt;span style="background-color: initial;"&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rational_number" rel="nofollow"&gt;rational numbers&lt;/a&gt;&lt;/span&gt;) between the frequencies of pitches. &lt;br /&gt;
In languages other than English, the original conceptions of &amp;quot;Just Intonation&amp;quot; are more obviously retained in the terms used in those languages: Reine Stimmung (pure, that is, beatless, tuning) in German, Натуральний стрій in Ukrainian and Gamme naturelle in French, (both referring to the &amp;quot;natural gamut&amp;quot;, that is, intervals derived from the harmonic partials), Intonazione naturale (natural intonation, once again intervals derived harmonic partials) in Italian, and so on. &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Just Intonation is sometimes distinguished from &lt;em&gt;rational intonation,&lt;/em&gt; by requiring that the ratios be ones of low complexity (as for example measured by &lt;a class="wiki_link" href="/Tenney%20height"&gt;Tenney height&lt;/a&gt;) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice, it can hardly be used except as a rank two 7-limit &lt;a class="wiki_link" href="/Microtempering"&gt;microtempering&lt;/a&gt; system because of certain very accurate approximations to the octave and to seven limit intervals: (6/5)^2/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)^9 is less than a cent short of an octave, while (27/25)^2 is almost precisely 7/6, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_minor_third" rel="nofollow"&gt;septimal minor third&lt;/a&gt;.&lt;br /&gt;
In the English language, the term &amp;quot;just&amp;quot; referred to &amp;quot;true, correct&amp;quot;, and is still used today in this sense, in the crafts. To &amp;quot;justify&amp;quot; a line of type is to fit it precisely to a certain measure, for example. The original sense, then, was similar to that sense which is clearly retained in other languages as &amp;quot;natural&amp;quot;. &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
If you are used to speaking only in note names, you may need to study the relation between frequency and &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pitch_%28music%29" rel="nofollow"&gt;pitch&lt;/a&gt;. Kyle Gann's &lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.kylegann.com/tuning.html" rel="nofollow"&gt;Just Intonation Explained&lt;/a&gt;&lt;/em&gt; is one good reference. A transparent illustration and one of just intonation's acoustic bases is the &lt;a class="wiki_link" href="/OverToneSeries"&gt;harmonic series&lt;/a&gt;.&lt;br /&gt;
If you are used to speaking only in note names, you may need to study the relation between frequency and &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pitch_%28music%29" rel="nofollow"&gt;pitch&lt;/a&gt;. Kyle Gann's &lt;em&gt;&lt;a class="wiki_link_ext" href="http://www.kylegann.com/tuning.html" rel="nofollow"&gt;Just Intonation Explained&lt;/a&gt;&lt;/em&gt; is one good reference. A transparent illustration and one of just intonation's acoustic bases is the &lt;a class="wiki_link" href="/OverToneSeries"&gt;harmonic series&lt;/a&gt;.&lt;br /&gt;
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